1. **Stating the problem:** Solve the system of linear equations: $$\begin{cases} x + y - z = 2 \\ 2x - y + z = 1 \\ x - y + 2z = 2 \end{cases}$$ 2. **Method:** We will use the substitution or elimination method to find $x$, $y$, and $z$. 3. **Step 1: Add equations (1) and (2) to eliminate $y$:** $$ (x + y - z) + (2x - y + z) = 2 + 1 $$ $$ x + y - z + 2x - y + z = 3 $$ $$ 3x = 3 $$ $$ x = 1 $$ 4. **Step 2: Substitute $x=1$ into equation (1):** $$ 1 + y - z = 2 $$ $$ y - z = 1 $$ 5. **Step 3: Substitute $x=1$ into equation (3):** $$ 1 - y + 2z = 2 $$ $$ -y + 2z = 1 $$ 6. **Step 4: Solve the system for $y$ and $z$:** $$ \begin{cases} y - z = 1 \\ -y + 2z = 1 \end{cases}$$ Add the two equations: $$ (y - z) + (-y + 2z) = 1 + 1 $$ $$ y - z - y + 2z = 2 $$ $$ z = 2 $$ 7. **Step 5: Substitute $z=2$ into $y - z = 1$:** $$ y - 2 = 1 $$ $$ y = 3 $$ 8. **Final answer:** $$ x = 1, \quad y = 3, \quad z = 2 $$